1. (General equilibrium with labor taxes) Robinson Crusoe lives for two periods, t = 1,2. He is endowed with coconuts in each period exogenousl'y, Y1 and Yg. In addition he can stores B coconuts at time 1 (at a neighboring island), which yields (1 + R)B coconuts at time 2.1 Then his budget constraint in each period is given by Q+B=n. (n Q=n+u+ma m He discounts future consumption by ,3 and his utility takes the log form. Then his utility maximization problem is given by $33.9 log Cl + [3 log Cg subject to the budget constraints (1) and (2) . (a) What are endogenous and exogenous variable or parameters? (b) Derive the intertemporal budget constraint. (Hint: present value of consumption 2 present value of income) (c) Derive the consumption Euler equation. (d) Solve for the optimal level of consumption and saving (That is, nd Ci\2. (Malthus Model with productivity shocks) Thomas Malthus, in 1798, argued that while population grew exponentially= the food supply increased only arithmetically= so that humanity was doomed to exceed its own means of subsistence. In order for the population to be placed back below its carrying capacity= Malthus belived that natural measures would be taken to lower the population such as disease, plague, and famine and that people should abstain from marriage to keep the fertility rate down. We formalize this idea. Consider an economy devoted to agriculture only. Output (Yr) is produced with productivity (At), labor (Ni) and land (D), and we suppose that the amount of land is xed and equal to l= D = 1. Then the relationship between inputs and output is described as follows: Yr : AtrtlOz' The population growth (23:1 ) is given by 2+1 = 'Nt 103 7 where \"Lot is wage at time t, wg population (Ni). is subsistence wage. Notice that labor is assumed to be equal to a) Firms determine their demand for labor through a lens of prot maximiation. Write down the condition for optimal labor demand by rms. b) Derive the law of motion for population (Hint: Plug labor demand in (a) into the population growth equation to get rid of wt, and express Nt+1 (lefthand side) in terms of N: and other variables (righthand side). c) Suppose that At = 1 for all t. Find out the steady state of the population using the law of motion for population from b). d) Suppose now that the discovery of a new ferilizer improves A: from 1 to 2. Following this change: will the economy expand? Why? Explain what will happen to the population Ni and wage to: over time